Two-Numbers and Their Applications-A Survey

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Remark 1.1. A maximal antipodal set A2M of a Riemannian manifold M is also later known as a 2-set or a great set for short in some literatures.

2. Antipodal points in algebraic topology Besides in geometry, the notion of antipodal points plays some signiﬁcant roles in many areas of mathematics, physics, and applied sciences. The following result from algebraic topology is well-known.

The Borsuk–Ulam Antipodal Theorem. Every continuous function from an n-sphere Sn into the Euclidean k-space Ek with k n maps some ≤ pair of antipodal points to the same point. In other word, if f : Sn Ek is → continuous, then there exists an x Sn such that f( x)= f(x). ∈ − Clearly, Borsuk-Ulam’s theorem fails for k>n, because Sn embedded in En+1. The Borsuk-Ulam theorem has numerous applications; range from combinatorics to diﬀerential equations and even economics. For n = 1, Borsuk–Ulam’s theorem implies that that at any moment there always exist a pair of opposite points on the earth’s equator with the same temperature. This assumes the temperature varies continuously. For n = 2, Borsuk–Ulam’s theorem implies that there is always a pair of antipodal points on the Earth’s surface with equal temperatures and equal barometric pressures, at any moment. TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY 3

The Borsuk–Ulam theorem was conjectured by S. Ulam at the Scottish Caf´ein Lvov, Ukraine. Ulam’s conjecture was solved in 1933 by K. Borsuk [8]. It turned out that the result had been proved three years before in [55] by L. Lusternik and L. Schnirelmann (cf. [8, footnote, page 190]). Since then many alternative proofs and also many extensions of Borsuk–Ulam’s theorem have been found as collected by H. Steinlein in his survey article [74] including a list of 457 publications involving various generalizations of Borsuk-Ulam’s theorem. Another important theorem from algebraic topology with the same ﬂavor is the following.

Brouwer’s Fixed-Point Theorem. Every continuous function f : B B → from the unit n-ball B into itself has a ﬁxed point.

Brouwer’s fixed-point theorem has numerous applications to many fields as well. For instance, Brouwer’s fixed-point theorem and its extension by S. Kakutani in [47] (extending Brouwer’s fixed-point theorem to set-valued functions) played a central role in the proof of existence of general equilib- rium in Market Economies as developed in the 1950s by economics Nobel prize winners K. Arrow (1972) and G. Debreu (1983).

Remark 2.1. In fact, Borsuk-Ulam’s antipodal theorem implies Brouwer’s ﬁxed point theorem, see [75].

3. Symmetric spaces The class of Riemannian manifolds with parallel Riemannian curvature tensor, i.e., R = 0, was first introduced independently by P. A. Shirokov ∇ in 1925 and by H. Levy in 1926. This class is known today as the class of locally symmetric Riemannian spaces (see, e.g., [16, page 292]). It was E.´ Cartan who noticed in 1926 that irreducible spaces of this type are separated into ten large classes each of which depends on one or two arbitrary integers, and in addition there exist twelve special classes corresponding to the exceptional simple groups. Based on this, E.´ Cartan created his theory of symmetric Riemannian spaces in his famous papers “Sur une classe remarquable d’espaces de Riemann” in 1926 [12]. Symmetric spaces are the most beautiful and important Riemannian manifolds. Such spaces arise in a wide variety of situations in both mathematics and physics. This class of spaces contains many prominent examples which are of great importance for various branches of mathematics, like compact Lie groups, Grassmannians and bounded symmetric domains. Symmetric 4 B.-Y. CHEN spaces are also important objects of study in representation theory, har- monic analysis as well as in differential geometry. An isometry s of a Riemannian manifold is called an involutive if s2 = id. A Riemannian manifold M is called a symmetric space if for each point x M there is an involutive isometry sx such that x is an isolated fixed ∈ point of sx; the involutive isometry sx = id is called the symmetry at x. 6 A Hermitian symmetric space is a Hermitian manifold M such that every point of M admits a symmetry preserving the Hermitian structure of M. Let M be a symmetric space. Denote by G the closure of the group of isometries on M generated by sp : p M in the compact-open topology. { ∈ } Then G is a Lie group which acts transitively on the symmetric space; hence the typical isotropy subgroup K, say at a point o M, is compact and ∈ M = G/K. Thus symmetric spaces are homogeneous spaces as well. From the point of view of Lie theory, a symmetric space is the quotient G/K of Lie group G by a Lie subgroup K, where the Lie algebra k of K is also required to be the (+1)-eigenspace of an involution of the Lie algebra g of G. In this paper, we will use standard symbols as in Helgason’s book [39] to denote symmetric spaces mostly. Here are a few minor exceptions. More specifically than AI, AI(n) denote SU(n)/SO(n), AII(n) := SU(2n)/Sp(n), n n n etc. Let Gd(R ), Gd(C ) and Gd(H ) denote the Grassmannians of d-dimensional subspaces in the real, complex and quaternion vector spaces (or modules), respectively. The standard notations for the exceptional spaces such as G2, F4, E6, ..., GI, ..., EIX denote the simply-connected spaces where we write GI for G2/SO(4). We will denote by M ∗ the bottom space, i.e., the adjoint space in [39], of the space M. For a symmetric space M, the dimension of a maximal flat totally geodesic submanifold of M is a well-defined natural number; called the rank of M and denoted by rk(M). Clearly, the rank of a symmetric space is at least one. It is well-known that the class of rank one compact symmetric spaces consists of n-sphere Sn, a projective space FP n(F = R, C, H) and the 16-dimensional Cayley plane F II = OP 2 with O being the Cayley algebra. Obviously, every complete totally geodesic submanifold of a symmetric space is also a symmetric space. It follows from the equation of Gauss that

rk(B) rk(M) (3.1) ≤ for every complete totally geodesic submanifold B of a symmetric space M (cf. [13]). TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY 5

4. (M+, M )–theory − In this section, we provide a brief introduction of the (M+, M )–theory − for compact symmetric spaces introduced by B.-Y. Chen and T. Nagano in [20, 21, 23, 62, 63]. Our approach to compact symmetric spaces based on antipodal points and fixed point sets of compact symmetric spaces. Hence our approach to compact symmetric spaces is quite different from other approaches done by E.´ Cartan and others. In fact, our approach to compact symmetric spaces plays the Key Roles in the determination of two-numbers of compact symmetric spaces; including the determination of 2-ranks of compact Lie groups (see [14, 17]). Let o be a point of a compact symmetric space M = G/K. A connected component of the fixed point set F (so, M)r o of the symmetry so is called { } a polar of o. We denote it by M+ or M+(p) if M+ contains a point p. We have the following useful propositions from [21] (also [14, page 15]). Proposition 4.1. Let M = G/K be a compact symmetric space. Then for each antipodal point p of o M, the isotropy subgroup K at o acts ∈ transitively on the polar M+(p). Moreover, we have K(p) = M+(p) and K(p) is connected. Consequently, we have M+(p) = K/Kp, where Kp is given by k K : k(p)= p . { ∈ } When a polar consists of a single point, we call it a pole. Proposition 4.2. Under the hypothesis of Proposition 4.1, the normal space to M+(p) at p in M is the tangent space of a connected complete totally geodesic submanifold M (p). Thus we have − dim M+(p)+dim M (p) = dim M. (4.1) − Proposition 4.3. For each antipodal point p of o in a compact symmetric space M, we have (1) rk(M (p)) = rk(M) and (2) M (p) is a connected − − component of the fixed point set F (sp so, M) of sp so through p. ◦ ◦ Polars and meridians of a compact symmetric space are totally geodesic submanifolds; they are thus compact symmetric spaces as well. Both polars and meridians have been determined for every compact connected irreducible Riemannian symmetric space (see [14, 23, 62, 63]). One of the most important properties of polars and meridians is that M is determined (globally) by any pair of (M+(p), M (p)). − Let o be a point of a compact connected Riemannian symmetric space M. If there exists a pole p of o in M, then the set consisting of the midpoints of all geodesics joining o and p is called the centrosome of o, p ; denoted by { } C(o, p). Every connected component of C(o, p) is again a totally geodesic 6 B.-Y. CHEN submanifold of M. Centrosomes play important roles in topology as well. For instance, centrosomes have been used by J. M. Burns in [11] to compute homotopy in many compact symmetric spaces (see subsection 6.4). The following result from [23, page 277] characterizes poles in compact symmetric spaces (see also [15]).

Proposition 4.4. The following six conditions are equivalent to each other for two distinct points o, p of a connected compact symmetric space M = GM /KG. (i) p is a pole of o M; ∈ (ii) sp = so; (iii) p is a polar of o M; { } ∈ (iv) there is a double covering totally geodesic immersion π = π o,p : { } M M with π(p)= π(o); → ′′ (v) p is a point in the orbit F (σ, GM )(o) of the group F (σ, GM ) through o, where σ = ad(so); (vi) the isotropy subgroup of SGM at p is that, SKG (of SGM at o), where SGM is the group generated by GM and the symmetries; SGM /GM is a group of order 2. ≤ For a compact symmetric space M, the Cartan quadratic morphism

Q = Qo : M GM → carries a point x M into sxso GM . The Cartan quadratic morphism is ∈ ∈ a GM -equivariant morphism which is an immersion. We have the following result for centrosomes from [23, pages 279-280].

Proposition 4.5. The following ﬁve conditions are equivalent to each other for two distinct points o, q of a connected compact symmetric space M.

(i) sosq = sqso; 2 (ii) Q(q) = 1GM , where Q = Qo is Cartan quadratic morphism; (iii) either so ﬁxes q or q is a point in the centrosome C(o, p) for some pole p of o; (iv) either so(q) = q or so(q) = γ(q) for the covering transformation γ for some pole p = γ(o) of o; (v) either so(q)= q or there is a double covering morphism π : M M → ′′ such that so′′ ﬁxes q′′, where o′′ = π(o) and q′′ = π(q).

The polars, meridians and centrosomes play very important roles in the study of compact connected symmetric spaces as well as of compact connected Lie groups. In particular, Propositions 4.4 and 4.5 play important TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY 7 roles for the study of maximal antipodal sets and two-numbers of compact symmetric spaces as well as the 2-ranks of compact Lie groups. Professor T. Nagano and M. Sumi (= M. S. Tanaka) proved in [64] that, for a compact symmetric space M, the root system R(M ) of a meridian − M = M is obtained from the Dynkin diagram of the root system R(M) − 6 of M and they also found all maximal totally geodesic spheres in SU(n) by means of the (M+, M )-method. − 5. Two-numbers of compact symmetric spaces 5.1. Antipodal set in terms of symmetries. For compact symmetric spaces, we have the following result form [23, page 275].

Proposition 5.1. For any compact symmetric space M, the two-number #2M is equal to the maximal possible cardinality #(A2M) of a subset A2M of M such that the point symmetry sx ﬁxes every point of A2M for every x A M. ∈ 2 For compact symmetric spaces, Proposition 5.1 can be regarded as an alternative deﬁnition of two-numbers. The following inequality from [23, page 276] is an important result for two-numbers.

Theorem 5.1. # M 1 does not exceed the sum of the two-numbers of all 2 − the polars of a point in a compact symmetric space M, that is # M 1+ # M . (5.1) 2 ≤ 2 + Remark 5.1. The equality (5.1) holdsX in many cases (such as the groups Sp(n) and O(n) and the hermitian symmetric spaces) and does not in the other cases (such as the adjoint group SU(8)∗ of SU(8)). 5.2. Two-numbers as an obstruction to totally geodesic embeddings. If B is a totally geodesic submanifold of a Riemannian manifold M, then every geodesic of B is also a geodesic of M. Thus we have # B # M. (5.2) 2 ≤ 2 Consequently, the invariant #2M is a geometric obstruction to the existence of a totally geodesic embeddings f : M N, since the existence of f clearly → implies the inequality # M # N. (5.3) 2 ≤ 2 For instance, although the real projective n-space RP n can be topolog- ically embedded in Sm for suﬃcient high m, but inequality (5.3) implies 8 B.-Y. CHEN that every real projective k-space RP k with k 2 cannot totally geodesic ≥ embedded into Sm regardless of codimension. This is simply due to the fact: k m #2RP = k + 1 > 2=#2S . 4 Similarly, while the complex Grassmann manifold G2(C ) of the 2-dimensional subspaces of the complex vector space C4 is obviously embedded into 6 4 G3(C ) as a totally submanifold, the space G2(C )∗ which one obtains by 4 4 identifying every member of G2(C ) with its orthogonal complement in C , 6 however, cannot be totally geodetically embedded into G3(C )∗ simply due to 4 6 #2G2(C )∗ = 15 > 12 = G3(C )∗ according to (5.1).

5.3. Two-numbers of dot products. Suppose a ﬁnite group Γ is acting on two spaces M and N freely as automorphism groups. Then Γ acts on the product space M N freely. And the orbit space (M N)/Γ is called × × the dot product of M and N (with respect to Γ) and denoted by M N. In · most cases of our study, Γ will be the group of order two acting on M and N as the covering transformation groups for double covering morphisms in the sequel. Γ will not be mentioned in that case, if Γ is obvious or if Γ need not be speciﬁed.

Example 5.1. SO(4) = S3 S3, U(n) = T SU(n) and GI has the only · · polar S2 S2. Here Γ for U(n) is the center of SU(n), a cyclic group of · order n.

The following result from [23, page 281] provides simple links between dot products, centrosomes and two-numbers of compact symmetric spaces.

Theorem 5.2. The dot product for double coverings M M and N N → ′′ → ′′ has the following properties: (a) # M # (M N); 2 ≤ 2 · (b) # (M N) # (F (so, M) F (sp,N))+# (CM CN), where CM 2 · ≤ 2 · 2 · and CN are the centrosomes for the point o of M and its pole and for pof N and its pole; (c) 1 (# M)(# N) # (M N) 2(# M )(# N ). 2 2 2 ≤ 2 · ≤ 2 ′′ 2 ′′ The following corollaries are easy consequences of Theorem 5.2.

Corollary 5.1. For a compact symmetric space M, we have n n 1 # (S M) # M +# (S − CM), 2 · ≤ 2 2 · where CM is a centrosome of M. TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY 9

Corollary 5.2. For a compact symmetric space M, we have # M # (S1 M) # M +# CM 2# M. 2 ≤ 2 · ≤ 2 2 ≤ 2 Corollary 5.3. We have # (Sm Sn) = 2(n + 1) if m n. 2 · ≥ 5.4. Two-numbers of irreducible compact symmetric spaces. In [23], Nagano and I have determined the two-numbers and the maximal antipodal sets for most irreducible compact symmetric spaces (see also Appendices II and III of [14, pages 67–71]). However, the antipodal sets of oriented real Grassmannians have not been discussed in [23]. On the other hand, H. Tasaki had described the maximal antipodal sets of oriented real Grassmannians in [89, 90].

6. Two-numbers and Topology The two-numbers link closely with topology. In this section, we shall present a number of results in this respect. 6.1. Two-numbers and Euler numbers. First, we present some links between two-numbers and Euler numbers of compact symmetric spaces from [23, Theorem 4.1]. Theorem 6.1. For any compact symmetric space M, we have # M χ(M), (6.1) 2 ≥ where χ(M) denotes the Euler number of M. The proof of this theorem based on a result of Hopf-Samelson on rank of G and K of a homogeneous space G/K with χ(G/K), a result of H. Hopf on ﬁxed point sets as well as the (M+, M )-theory of symmetric spaces. − For any hermitian compact symmetric space of semisimple type, Professor Nagano and I proved in [23, Theorem 4.3] the following. Theorem 6.2. For any compact hermitian symmetric space M of semisimple type, we have

#2M = χ(M)=1+ #2M+. (6.2)

The proof of this theorem based heavilyX on the (M+, M )–theory as well − as the Lefschetz ﬁxed point theorem in the version of Atiyah and Singer [5].

An immediate consequence of this theorem is the following. Corollary 6.1. For every complete totally geodesic hermitian subspace B of a semi-simple hermitian symmetric space M, we have χ(M) χ(B). (6.3) ≥ 10 B.-Y. CHEN

6.2. Two-numbers and homology. A symmetric R-space is a special kind of compact symmetric space for which several characterizations were known. Originally, Professor T. Nagano introduced the notion of a symmetric R- space in [61] as a compact symmetric space M which admits a Lie transformation group P which is non-compact and contains I0(M) as a subgroup. For example, a sphere is a symmetric R-space. The I(M) of M = Sn coincides with the natural action of O(n+1) on Sn. Thus the action of SO(n+1) n on S coincides with I0(M). In the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup. Subgroups between a Borel subgroup B and the ambient group G are called parabolic subgroups. Working over algebraically closed ﬁelds, the Borel subgroups turn out to be the minimal parabolic subgroups in this sense. Thus B is a Borel subgroup when the homogeneous space G/B is a complete variety which is as large as possible. M. Takeuchi used the terminology symmetric R-space for the ﬁrst time in [77]. He gave a cell decomposition of an R-space in [77], which is a kind of generalization of a symmetric R-space. Here, by a R-space we mean M = G/U where G is a connected real semisimple Lie group without center and U is a parabolic subgroup of G. A compact symmetric space M is said to have a cubic lattice if a maximal torus of M is isometric to the quotient of Er by a lattice of Er generated by an orthogonal basis of the same length. O. Loos [53] gave another intrinsic characterization of symmetric R-spaces among all compact symmetric spaces with the property that the unit lattice of the maximal torus of the compact symmetric space (with respect to a canonical metric) is a cubic lattice. Loos’ proof was based on the correspondence between the symmetric R-spaces and compact Jordan triple systems. In [35], D. Ferus characterized symmetric R-spaces as compact symmetric submanifolds of Euclidean spaces. He also proved that compact extrinsically symmetric submanifolds are orbits of isotropy representations of symmetric space of compact type (or non-compact type).

The class of symmetric R-spaces includes:

(a) all hermitian symmetric spaces of compact type (b) Grassmann manifolds O(p+q)/O(p) O(q),Sp(p+q)/Sp(p) Sp(q) × × (c) the classical groups SO(m),U(m), Sp(m), (d) U(2m)/Sp(m),U(m)/O(m), (e) (SO(p + 1) SO(q + 1))/S(O(p) O(q)), where S(O(p) O(q)) is × × × the subgroup of SO(p + 1) SO(q + 1) consisting of matrices of the × TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY 11

form: ǫ 0 0 A   , ǫ = 1, A O(p), B O(q), ǫ 0 ± ∈ ∈    0 B     (This R-space is cover twice by Sp Sq), × (f) the Cayley projective plane FII = OP 2, and (g) the three exceptional spaces E /Spin(10) T, E /E T, and E /F . 6 × 7 6× 6 4 For any symmetric R-space, M. Takeuchi proved in [79] the following.

Theorem 6.3. For any symmetric R-space M, we have

#2M = bi(M, Z2), (6.4) i 0 X≥ where bi(M, Z2) is the i-th Betti number of M with coeﬃcients in Z2.

This theorem was proved by applying a result of Chen-Nagano from [23] in conjunction with an earlier result of M. Takeuchi from [77].

6.3. Bott’s periodicity theorem for homotopy of classical groups. The most famous work of R. Bott is his periodicity theorem which describes periodicity in the homotopy groups of classical groups (cf. [10]). Bott’s original results may be succinctly summarized as

Theorem 6.4. The homotopy groups of the classical groups are periodic:

πi(U)= πi+2(U), πi(O)= πi+4(Sp), πi(Sp)= πi+4(O) for i = 0, 1, , where U is the direct limit deﬁned by U = U(k) and · · · ∪k∞=1 similarly for O and Sp.

The second and third of these isomorphisms given in Theorem 6.4 yield the 8-fold periodicity:

πi(O)= πi+8(O), πi(Sp)= πi+8(Sp) for i = 0, 1, . · · · Bott’s original proof of his periodicity theorem is diﬀerential geometric in its nature (see [9]). His proof relies on the observation that in a compact Riemannian symmetric space M one can choose two points p and q in “special position” such that the connected components of the space of shortest geodesics in M joining p and q are again compact symmetric spaces. Put M = M0 and let M1 be one of the resulting connected components. This construction can be repeated inductively. 12 B.-Y. CHEN

Let M be a compact symmetric space. The index of a geodesic γ in M from p to q is the number of conjugate points of p counted with their multiplicities, in the open geodesic segment from p to q. We denote the space of shortest geodesics from p to q by the symbol Ωd, which relates closely with the notion of centrosomes. The proof of Bott’s periodicity theorem in [9] relied on the following result.

Rott’s theorem. If Ωd is a topological manifold and if every non-shortest geodesic from p to q has index greater than or equal to λ0, then the (i+1)-th homotopy groups of M satisfies d πi+1(M) ∼= πi(Ω ) (6.5) for i < λ 1. 0 − 6.4. Applications to homotopy. The (M+, M )–theory as well as centro- − somes play significant roles in computing homotopy of compact symmetric spaces. In fact, J. M. Burns mentioned in the introduction of his paper [11] that how our (M+, M )–theory (in particular, centrosomes) in conjunction with − Bott’s theorem given above can be used to compute the homotopy of compact symmetric spaces. In [11], J. M. Burns carried out the computation of the homotopy in compact symmetric spaces of types: AI, AII and CI. By applying the same method, he also computed in [11] the homotopy in the exceptional symmetric spaces: EIII EIX, F , FI and F II. − 4 The homotopy groups πi(EVII) with i 16 were computed in 2012 by ≤ P. Quast [67]. The first eight homotopy groups of EVII have been already determined by J. M. Burns in [11].

Remark 6.1. It is known that the homotopy group πi(M) of a connected manifold M is the same as the homotopy group πi(M˜ ) of its universal cover M˜ for i 2. Also, the homotopy groups π ,π and π of EVII can also be ≥ 9 10 14 directly read oﬀ from the long exact sequence of homotopy groups of coset spaces together with the corresponding homotopy groups of E6 and E7 that can be found in [59, page 363]. 6.5. Two-numbers and covering maps. In [23], Professor T. Nagano and I also discovered some simple links between two-numbers and covering maps between compact symmetric spaces. For double coverings we have [23, page 280]: Theorem 6.5. If M is a double covering of M , then # M 2 # (M ). ′′ 2 ≤ · 2 ′′ Remark 6.2. Th inequality in Theorem 6.5 is sharp, since the equality case holds for M = SO(2m) with m> 2. TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY 13

For k-fold coverings with odd k, [23, page 278]: Theorem 6.6. Let φ : M N is a k-fold covering between compact sym- → metric spaces. If k is odd then #2M =#2(N). 6.6. Antipodal sets, 2-numbers and Borsuk-Ulam’s type theorems. A continuous function f : M R of a compact symmetric space M = G/K → into the real line is called isotropic if f is invariant under the action of the isotropic subgroup K. Recently, I obtained the following Borsuk-Ulam’s type results involving antipodal sets and two-numbers of compact symmetric spaces in [18]. Theorem 6.7. Let f : M R be a continuous isotropic function from a → compact symmetric space into the real line. Then f carries some maximal antipodal set of M to the same point in R, whenever M is one of the following spaces: Spheres; the projective spaces FP n (F = R, C, H); the Cayley plane F II; the exceptional spaces EIV ; EIV ∗; GI; and the exceptional Lie group G . Consequently, the function f : M R carries a maximal antipodal set 2 → with #2M elements to the same point in R. All of the compact symmetric spaces M mentioned in the list of Theorem 6.7 have single polar for a ﬁxed point o M. Next, we discuss the case in ∈ which the point o M has multiple polars. Suppose o M has multiple 1 2∈ k ˆ ∈ polars, say M+, M+,...,M+. Then we denote by M+ a polar of o with the maximal two-number among all polars of o. We also have the following results from [18]. Theorem 6.8. Let f : M R be a continuous isotropic function of a → compact symmetric space M into the real line R. If M admits more than one polar, then f carries an antipodal set SM of M consisting of 1+#2Mˆ + points of M to the same point in R. The following corollaries are easy consequences of Theorem 6.7 (see [18] for more details). Corollary 6.2. If f : E R is a continuous isotropic function of the 8 → exceptional Lie group E8 into the real line R, then f carries an antipodal set of E8 with 392 elements to the same point in R. Corollary 6.3. If f : FI R is a continuous isotropic function of the → compact symmetric space FI into the real line R, then f carries an antipodal set of FI with 24 elements to the same point in R. We provide a simple example to illustrate that the isotropic condition on f in Theorem 6.7 is essential. 14 B.-Y. CHEN

Example 6.1. Consider the real projective plane RP 2. Then there is a double covering π : S2 RP 2. Without loss of generality, we may assume → that S2 is the unit sphere in E3 centered at the origin of E3. For each real- valued function f : RP 2 R, the lift fˆ : S2 R of f is an even function → → via the double covering π, i.e., fˆ( x)= fˆ(x), x = (x,y,z) S2. − ∀ ∈ Conversely, for every given real-valued even function g : S2 R of S2, g → induces a real-valued functiong ˇ : RP 2 R of RP 2. → If we choose g : S2 R to be g(x) = (x y)2, then g induces a real-valued → − functiong ˇ : RP 2 R which does not carry any maximal antipodal set of → RP 2 to the same point in R.

7. 2-rank of compact connected Lie groups 7.1. 2-ranks of Borel and Serre. A. Borel and J.-P. Serre considered in [7] the 2-rank, denoted by r2G, of a compact connected Lie group G. By deﬁnition the 2-rank of a compact connect Lie group G is the maximal possible rank of the elementary 2-subgroup of G. In [7] Borel and Serre proved the following 2 results. (1) rk(G) r (G) 2rk(G) with rk(G)= rank(G) and ≤ 2 ≤ (2) G has (topological) 2-torsion if rk(G) < r2(G). Borel and Serre were able to determine the 2-rank of simply-connected simple Lie groups SO(n),Sp(n),U(n), G2 and F4 in [7]. Furthermore, they proved that the exceptional Lie groups G2, F4 and E8 have 2-torsion. On the other hand, they also mentioned in [7, page 139] that they were unable to determine the 2-rank for the exceptional simple Lie groups E6 and E7.

7.2. 2-rank and commutative algebra. After Borel and Serre’s paper [7], 2-ranks of compact Lie groups G have been investigated by a number of mathematicians. In particular, it had been shown that the 2-ranks of compact Lie groups have several important links with commutative algebra. Here, we provide merely two such links.

(a) Let F be either a ﬁeld or the rational integer ring Z. Let A = i 0 Ai be a graded commutative F -algebra in sense of Milnor-Moore [60]. If≥A is P connected, then it admits a unique augmentation ε : A F . → Put A¯ = Ker ε. The A¯ is called the augmentation ideal of A. A sequence of elements x1,...,xn A¯ in A¯ is called a simple system of generators if ǫ1 ǫn { ∈ } x x : ǫi = 0 or 1 is a module base of A. { 1 · · · n } Let G be a compact connected Lie group. Denote by s(G) the number of generators of a simple system of the Z2-cohomology H∗(G, Z2) of G. TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY 15

A. Kono discovered in [50] the following relationship between s(G), r2G and the Z2-cohomology H∗(G, Z2).

Theorem 7.1. Let G be a connected compact Lie group. Then the following three conditions are equivalent: (1) s(G) r G; ≤ 2 (2) s(G)= r2G; (3) H∗(G, Z2) is generated by universally transgressive elements.

To prove Theorem 7.1, A. Kono had applied May’s spectral sequence [58], Eilenberg-Moore’s spectral sequence [33] as well as Quillen’s result in [69]. In [50], Kano also described some properties of compact Lie groups satisfying condition (3) in Theorem 7.1 and provided applications. (b) In commutative algebra, Krull’s dimension of a ring R is the supremum of the number of strict inclusions in a chain of prime ideals. More precisely, one says that a strict chain of inclusions of prime ideals of the form:

p ( p ( ( pn 0 1 · · · is of length n. That is, it is counting the number of strict inclusions. Given a prime ideal p R, we deﬁne the height of p to be the supremum ⊂ of the set n N : p is the supremum of a strict chain of length n . Then { ∈ } the Krull dimension is the supremum of the heights of all of its primes.

Let G be a compact Lie group and put HG∗ = H∗(BG; Z2), where BG is a classifying space for G. Let NG∗ HG∗ denote the ideal of nilpotent elements. ⊂ # Then it is known that HG∗ /NG∗ = HG is a finitely generated commutative algebra. In [69], D. Quillen studied the relationship between the finitely generated # commutative algebra HG and the structure of the Lie group G. In particular, # he proved that, under some suitable assumptions, the Krull dimension of HG is equal to the 2-rank of G. D. Quillen proved his result by calculating the mod 2 cohomology ring of extra special 2-groups. Quillen’s result provided an affirmative answer to a conjecture of M. F. Atiyah [2] and also of a conjecture of R. G. Swan [76].

8. Applications of two-numbers to group theory

1 If G is a connected compact Lie group, then by assigning sx(y)= xy− x to 2 every point x G, we have s = idG to each point x. Therefore G becomes a ∈ x compact symmetric space with respect to a bi-invariant Riemannian metric. 16 B.-Y. CHEN

8.1. Links between two-numbers and 2-ranks. The following simple link between the 2-rank and the two-number of a connected compact Lie group G was established in [22, Proposition 5].

Theorem 8.1. Let G be a connected compact Lie group. Then we have

r2G #2G = 2 . (8.1) For products of two compact Lie groups, we have the following result from [23, Lemma 1.7].

Theorem 8.2. Let G1 and G2 be connected compact Lie groups. Then # (G G ) = 2r2G1+r2G2 . (8.2) 2 1 × 2 Based on these two theorems and (M+, M )–theory, we were able to − determine the 2-ranks of compact connected Lie groups in [23, pages 289- 293]. Consequently, we settled the problem of Borel-Serre for determination of 2-ranks of connected compact simple Lie groups as follows.

8.2. 2-ranks of classical groups. For classical groups we have:

Theorem 8.3. Let U(n)/Zµ by the quotient group of the unitary group U(n) by the cyclic normal subgroup Zµ of order µ. Then we have n + 1 if µ is even and n = 2 or 4; r2(U(n)/Zµ)= (8.3) (n otherwise. Theorem 8.4. For SU(n)/Zµ, we have

n + 1 for (n,µ) = (4, 2);

r2(SU(n)/Zµ)= n for (n,µ) = (2, 2) or (4, 4); (8.4)  n 1 for the other cases. − Theorem 8.5. One has r (SO(n)) = n 1 and, for SO(n) , we have 2  − ∗ 4 for n = 4; r2(SO(n)∗)= (8.5) (n 2 for n even > 4. − Theorem 8.6. Let O(n) = O(n)/ 1 . We have ∗ {± } (a) r2(O(n)) = n; (b) r (O(n) ) is n if n is 2 or 4, while it is n 1 otherwise. 2 ∗ − Theorem 8.7. One has r2(Sp(n)) = n, and, for Sp(n)∗, we have n + 2 for n = 2 or 4 r2(Sp(n)∗)= (8.6) (n + 1 otherwise. TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY 17

Thus we also have

r2(Sp(n)∗)= r2(U(n)/Z2) + 1 (8.7) for every n.

8.3. 2-ranks of spinors, semi-spinors and P in(n). Next, we consider the spinor Spin(n) and its related groups. Recall that Spin(n) is a subset of the Cliﬀord algebra Cl(n) which is generated over R by the vectors ei in n the ﬁxed orthonormal basis of R ; subject to the conditions eiej = ejei − and eiei = 1, i = j. − 6 For Spin(n) we have the following two results.

Theorem 8.8. We have r + 1 if n 1, 0 or 1 (mod 8) r2(Spin(n)) = ≡− (r otherwise, n where r is the rank of Spin(n), r = [ 2 ]. Theorem 8.9. (PERIODICITY) For n 0, One has ≥ r2(Spin(n + 8)) = r2(Spin(n)) + 4

The group P in(n) was introduced in [4] by M. F. Atiyah, R. Bott and A. Shapiro while they studied Cliﬀord modules. P in(n) is a group in the Cliﬀord algebra Cl(n) and it double covers O(n) and whose connected component Spin(n) double covers SO(n). For P in(n) we have:

Theorem 8.10. For P in(n) we have r (P in(n)) = r (Spin(n + 1)), n 0. 2 2 ≥ For the semi-spinor group SO(4m)# = Spin(4m)/ 1, e , we have: { ((4m)) } Theorem 8.11. We have 3 if m = 1

# 6 if m = 2, r2(SO(4m) )=  r + 1 if m is even > 2,  r if m is odd > 1,   where r is the rank 2m of SO(4m)#.

Remark 8.1. The 2-rank of Spin(16) and of SO(16)# had been obtained independently by J. F. Adams in [1]. However, the method of his proof is completely diﬀerent from ours given in [23]. 18 B.-Y. CHEN

8.4. 2-ranks of exceptional groups. We also able to determine the 2- ranks of Exceptional groups as follows.

Theorem 8.12. One has

r2G2 = 3, r2F4 = 5, r2E6 = 6, r2E7 = 7, r2E8 = 9 for the simply-connected exceptional simple Lie groups.

Remark 8.2. r2G2 =3 and r2F4 = 5 are due to A. Borel and J. P. Serre.

We also able to determine the rank of the bottom space E6∗.

Theorem 8.13. One has r2E6∗ = 6.

9. Two-numbers and algebraic geometry 9.1. Two-numbers and projective ranks. The projective rank, P r(M), of a compact hermitian symmetric space M is the maximal complex dimension of totally geodesic complex projective spaces N of M deﬁned by A. Fauntleroy in [34]. A subset E of a maximal antipodal set of a compact symmetric space M is called equidistant if there is a real number a > 0 such that dM (x,y) = a for any two points x,y E. ∈ Put

γ = γM = min dM (x,y) : x,y E,x = y (9.1) { ∈ 6 } and let Aγ E be an equidistance set (for the distance γ) of maximal ⊂ cardinality. Let µ = #(Aγ ) denote the cardinality of Aγ . C. U. S´anchez and A. Guinta proved in [72] the following.

Theorem 9.1. Let M be a compact connected irreducible hermitian symmetric space. (a) If M = CI(n), then P r(M)= µ 1. 6 − (b) If M = CI(n) and A√2γ is an equidistant set of maximal cardinality (for the distance √2γ(M)) and µ is its cardinality, then P r(M) = µ 1. − Theorem 9.1 provides the following simple link between the two-number and the projective rank for compact irreducible hermitian symmetric spaces.

Theorem 9.2. P r(M) rk(M) # (M) for every compact irreducible · ≤ 2 hermitian symmetric space M. TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY 19

9.2. Two-numbers and arithmetic distances. W.-L. Chow introduced in [26] the notion of arithmetic distance “d” for every classical hermitian symmetric spaces M. In case that the classical hermitian symmetric space n M is the complex Grassmannian Gp(C ), W.-L. Chow’s arithmetic distance n d(V,W ) between V,W Gp(C ) is deﬁned by ∈

d(V,W ) = dimC V/V W, (9.2) ∩ i.e., the codimension of their intersection V W in V (or in W ). Equivalently, ∩ the arithmetic distance d(V,W ) is the smallest integer t such that there n exists a ﬁnite set Ui 1 i t+1 of linear subspaces in Gp(C ) such that { } ≤ ≤ (a) U1 = V and Ut+1 = W ; (b) d(Ui,Ui )=1for 1 i t. +1 ≤ ≤ Chow proved in [26] that the d-preserving transformations of M are either holomorphic or anti-holomorphic, provided rk(M) > 1

A Helgason sphere in a compact symmetric space is a maximal-dimensional totally geodesic sphere with maximal sectional curvature. The dimension of a Helgason sphere is thus equal to one more than the multiplicity of the highest restricted root. Using Helgason spheres, S. Peterson [66] extended Chow’s arithmetic distance to an arithmetic distance d to arbitrary irreducible compact symmetric spaces. In this general case, d(x,y) satisfies d(x,y) j if the two points ≤ x,y M are joined by a chain of j Helgason spheres. ∈ More precisely, put d(x,y)=0if x = y. And put d(x,y)=1if x = y and 6 x,y lie in a Helgason sphere. Otherwise, d(x,y) is defined to be the smallest j such that there is a chain of j Helgason spheres joining x and y (instead n of a chain of linear subspaces in Gp(C )). If M is hermitian and irreducible, the holomorphic transformations per- mute the Helgason spheres and hence they preserve the new arithmetic dis- n tance obviously. Also, it equals Chow’s arithmetic distance if M = Gp(C ). Let M = G/K be a compact Riemannian symmetric space where G denotes the identity component of the isometry group of M. T. Nagano studied and classified in [61] the geometric transformation groups of compact symmetric spaces. Roughly speaking, they are “larger groups” L that act on M such that (i) G L; ⊂ (ii) L is a Lie transformation group acting effectively on M; (iii) L preserves the symmetric structure of M; and (iv) L is simple. 20 B.-Y. CHEN

By using our approach to compact symmetric spaces and a formulation of Radon’s duality, S. Peterson proved the following rigidity theorem in [66].

n Theorem 9.3. Let M be a Gd(F ), F = R, C, H, or M = AI(n) with dim M 3 and let L = ϕ : M M, ϕ a diﬀeomorphism preserving the ≥ { → arithmetic distance d . Then L = L , where L be the geometric transforma- } ′ ′ tion group of M.

10. Further applications of two-numbers 10.1. Real forms of hermitian symmetric spaces. Let M be a hermitian symmetric space of compact type and let τ be an involutive anti- holomorphic isometry of M. The ﬁxed point set F (τ, M)= x M : τ(x)= x { ∈ } is called a real form of M, which is connected and a totally geodesic La- grangian submanifold. For instance, let M = CP 1 = S2, L = RP 1 = S1. Then M is a hermitian symmetric space of compact type and L is a real form of M. Any two distinct great circles in S2 intersect at two points which are antipodal to each other. The intersection is an antipodal set in L as well as in M. M. Takeuchi proved the following result in [78].

Theorem 10.1. Every real form of a hermitian symmetric space of compact type is a symmetric R-space. Conversely, every symmetric R-space is realized as a real form of a hermitian symmetric space of compact type. Moreover, the correspondence is one-to-one.

In [84], M. S. Tanaka and H. Tasaki studied the intersection of two real forms in a hermitian symmetric space of compact type. In particular, they proved that the intersection of two real forms in a Hermitian symmetric space of compact type is an antipodal set if the intersection is discrete. More precisely, they established the following link between real forms of hermitian symmetric spaces and two-numbers in [84].

Theorem 10.2. Let L1,L2 be real forms of a hermitian symmetric space of compact type whose intersection is discrete. Then L L is an antipodal 1 ∩ 2 set in L and L . Furthermore, if L and L are congruent, then L L is 1 2 1 2 1 ∩ 2 a 2-set. Thus #(L L )=# L =# L . 1 ∩ 2 2 1 2 2 Further results in this respect see [44, 86, 87]. TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY 21

10.2. Fixed point set of a holomorphic isometry. O. Ikawa, M. S. Tanaka and H. Tasaki showed in [45] a necessary and sufficient condition that the fixed point set of a holomorphic isometry of a Hermitian symmetric space of compact type is discrete and proved that the discrete fixed point set is an antipodal set. They also proved a necessary and sufficient condition that the intersection of two real forms in a Hermitian symmetric space of compact type is discrete and consider some relations between the intersection of two real forms and the fixed point set of a certain holomorphic isometry by the use of the symmetric triads introduced by O. Ikawa in [43].

10.3. Applications to Lagrangian Floer homology. Let (M,ω) be a symplectic manifold, i.e., M is a manifold with a closed nondegenerate 2- 1 form ω. Let L be a Lagrangian submanifold in M, i.e., dimR L = 2 dimR M and ω vanishes on L. The symplectic Floer homology is a homology theory associated to a symplectic manifold and a nondegenerate symplectomorphism of it. If the symplectomorphism is Hamiltonian, the homology arises from studying the symplectic action functional on the (universal cover of the) free loop space of a symplectic manifold. Symplectic Floer homology is invariant under Hamiltonian isotropy of the symplectomorphism. Let Hamilt(M,ω) denote the set of all Hamiltonian diffeomorphisms of M. A. Floer defined in [36] the homology when π2(M,Li) = 0, i = 0, 1, and proved that it is isomorphic to the singular homology group H (L0, Z2) of L0 ∗ in the case where L0 is Hamiltonian isotopic to L1. As a result, Floer solved affirmatively the so called Arnold conjecture for Lagrangian intersections in that case. In [37], A. Givental posed the following conjecture which generalized the results of Floer and himself.

Conjecture (Arnold-Givental). Let (M,ω) be a symplectic manifold and τ : M M be an anti-symplectic involution of M. Assume that → the ﬁxed point set L = F (M,τ) is not empty and compact. Then for any φ Hamilt(M,ω) such that the Lagrangian submanifold L and its image ∈ φ(L) intersect transversally, the inequality

#(L φ(L)) b(L, Z ) (10.1) ∩ ≥ 2 holds, where b(L, Z2)= i 0 bi(L, Z2). ≥ Recall that M. TakeuchiP proved that if L is a symmetric R-space, then one has #2L = b(L, Z2) (cf. Theorem 6.3). H. Iriyeh, T. Sakai and H. 22 B.-Y. CHEN

Tasaki computed in [46] the Lagrangian Floer homology HF (L0,L1; Z2) of a pair of real forms (L0,L1) in a monotone hermitian symmetric space M of compact type in the case where L0 is not necessarily congruent to L1. In particular, they obtained a generalization of the Arnold-Givental inequality (10.1) in the case where M is irreducible. As an application, H. Iriyeh, T. Sakai and H. Tasaki proved the following. Theorem 10.3. Every totally geodesic Lagrangian sphere in the complex hyperquadric is globally volume minimizing under Hamiltonian deformations. 10.4. Two-numbers and tight Lagrangian submanifolds. The complex hyperquadric Qn(C) is holomorphically isometric to the compact her- n+2 mitian symmetric space G˜2(R ). For every natural number k with 0 k n, the complex hyperquadric k,n k ≤ ≤ k,n k k n k Qn(C) admits certain real forms S deﬁned by S = (S S )/Z . − − × − 2 H. Tasaki proved in [88] the following.

n Theorem 10.4. Let k,ℓ be integers with 0 k ℓ [ 2 ], and let L1 and L2 k,n ≤k ≤ ≤ℓ,n ℓ be real forms which are congruent to S − and S − , respectively. If L1 and L2 intersect transversally, then L1 L2 is a 2-set of L1 and an antipodal n ∩ set of L2. Moreover, if k = ℓ = [ 2 ], then the two-numbers satisfy # (G˜ (Rn+2)) = #(L L ). 2 2 1 ∩ 2 By applying Theorem 10.4, H. Tasaki obtained the following.

n+2 Theorem 10.5. Any real form of the oriented real Grassmannian G˜2(R ) is a globally tight Lagrangian submanifold. 10.5. Application to convexity. Let M be a Riemannian manifold and let τ be an involutive isometry of M. A connected component of the fixed point set of τ with positive dimension is called a reflective submanifold, which is a totally geodesic submanifold of M. A connected submanifold S of M is called (geodetically) convex if any shortest geodesic segment in S is still shortest in M. The (M+, M )–theory was also used in [68] by P. Quast and M. S. Tanaka − to prove the following result. Theorem 10.6. Every reflective submanifold of a symmetric R-space is convex.

10.6. Applications to stability. By applying the (M+, M )–theory for − compact symmetric spaces, the author, P.-F. Leung and T. Nagano introduced in 1980 an algorithm in [19] to determine the stability of totally geodesic submanifolds of compact symmetric spaces. TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY 23

In 1987, Y. Ohnita reformulated and improved the above algorithm in [65] to include the formulas for the index, the nullity and the Killing nullity of a compact totally geodesic submanifold in a compact symmetric space. In the same paper, Ohnita proved that every Helgason sphere in any compact symmetric space is stable. Subsequently, many results about the stability of totally geodesic submanifolds in compact symmetric spaces were obtained by applying our algorithm and Ohnita’s reformulation. In 1990, K. Mashimo and H. Tasaki proved in [57] that every closed Lie subgroup of Dykin index 1 in every compact simple connected Lie group is stable. K. Mashimo also determined in [56] all of unstable Cartan embeddings of compact symmetric spaces. Moreover, by applying the algorithm there are results on the stability of symmetric R-spaces in Hermitian symmetric spaces obtained by M. Takeuchi [78] and also the stability of polars and meridians by M. S. Tanaka in [80]. Furthermore, T. Kimura and M. S. Tanaka determined in [49] the stability of the maximal totally geodesic submanifolds in compact symmetric spaces of rank two by applying the algorithm.

10.7. Two-numbers and algebraic coding theory. Coding theory is the study of the properties of codes and their respective ﬁtness for speciﬁc applications. The main purpose of codes is to be able to recover the original content of a transmitted message by correcting errors that have entered the message during transmission. This capability is useful in maintaining the integrity of communication systems, computer networks, compact disk recording, etc. A p-group H is called extra special if its center Z is cyclic of order p, and the quotient H/Z is a non-trivial elementary abelian p-group. J. A. Wood explored in [91] the equivalence between the diagonal extra- special 2-group of the connected compact Lie group Spin(n) and the self- orthogonal linear binary codes of algebraic coding theory. Our results in subsection 8.3 (Theorems 8.8 and 8.9) were mentioned and been used in his work in [91].

10.8. An application to theory of designs. The theory of designs is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry. In 1973, P. Delsarte [28] unified the theories of codes and designs on association schemes and gave the upper bounds for codes and the lower bound 24 B.-Y. CHEN for designs by applying linear programming for polynomials associated with metric or cometric association schemes. After Delsarte’s work, the theory of spherical designs was introduced by Delsarte-Goethals-Seidel in [29] as an analogy of Delsarte technique. The essential tool in their works is the addition formula for polynomials; polynomials associated with metric or cometric association schemes, or the Gegen- bauer polynomials with spheres. In general, the theory of designs can be given on the Delsarte spaces or the polynomial spaces, which are metric spaces with “good” polynomials, such as the polynomials associated with metric or cometric association schemes or Gegenbauer’s polynomials. Compact symmetric spaces of rank one are natural and significant examples of the Delsarte spaces or the polynomial spaces for continuous metric spaces. The theory of designs on compact symmetric spaces of rank one was studied in details by S. G. Hoggar in [40]. For compact symmetric spaces of higher rank, H. Kurihara and T. Okuda proved in [52] a characterization of maximal antipodal sets of complex Grass- mannians in term of certain designs (more precisely, E F-designs) with the ∪ smallest cardinalities.

11. Extensions of two-numbers and applications 11.1. Holomorphic two-numbers of compact hermitian symmetric H spaces. C. U. Sánchez defined holomorphic two-number #2 (M) for a compact connected hermitian manifold M in [71] as the maximal possible cardinality of a subset A2 such that for every pair of points x and y of A2, there exists a totally geodesic complex curve of genus 0 in M on which x and y are antipodal to each other. C. U. Sánchez proved the following.

H Theorem 11.1. #2 M = #2(M) for every compact hermitian symmetric space.

By combining Theorem 11.1 with our equality #2M = χ(M) from The- orem 6.2, one has the following.

Corollary 11.1. [71] For every compact hermitian symmetric space M, one H has #2 (M) = χ(M). 11.2. Index numbers for real ﬂag manifolds. A real ﬂag manifold (also called a R-space) is a homogeneous space of the form G/P , where G is a real semisimple Lie group without compact factors and P is a parabolic subgroup. Here G/P K/K P which is a K-orbit on P . ≡ ∩ TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY 25

Every complex flag manifold may be considered an R-space. In fact, let U be a compact connected semisimple centerless Lie group and u its Lie algebra. The complex flag manifolds of U are the orbits of the adjoint action of U on u. Take M = Ad(U)Y for Y =0 in U. Let g = uc = u + iu. Then there is 6 a Cartan decomposition of the realization gR of g and one may consider M as the orbit of iY in iu by the adjoint action of U. In [71], C. U. Sánchez proved the following. Proposition 11.1. Let M be a real flag manifold. Then there exists a complex flag manifold Mc such that M is isometrically imbedded in Mc. If M is a symmetric R-space, then Mc is a hermitian symmetric space and the isometric imbedding is totally geodesic. If M is already a complex flag manifold, then Mc = M. Using this fact that every real flag manifold M can be isometrically embedded into a complex flag manifold MC (in fact, it is a connected component of an antisymplectic involution on a complex flag manifold), C. U. Sánchez defined in [71] the index number #I M of a real flag manifold M as the maximal possible cardinality of the p-sets ApM (for a prime p), defined in terms of fixed points of symmetries of the complex flag manifolds restricted to the real one. As an extension of Theorem 6.3 of M. Takeuchi, Sánchez proved the following result in [71].

Theorem 11.2. Let M be a real flag manifold. Then #I M = b(M, Z2), where b(M, Z2)= i 0 dim bi(M, Z2). ≥ 11.3. k-numbersP for complex flag manifolds. Now, we explain very briefly the notion of k-number for complex flag manifold introduced by Sánchez in [70]. For every complex flag manifold MC there exists a positive integer k0 = ko(MC) 2 such that for each integer k k there exists a k-symmetric ≥ ≥ 0 structure on MC, i.e., for each point x MC there exists an isometry θx k ∈ satisfying θx = id with x as an isolated fixed point. Following [51], a k-symmetric structure is called regular it satisfies

θx θy = θz θz (11.1) ◦ ◦ with z = θx(y). As an extension of two-numbers for symmetric spaces, C. U. Sánchez defined the k-number #k(MC) of the complex flag manifold MC as the maximal possible cardinality of the so-called k-sets Ak MC with the property that ⊂ for each x Ak the corresponding k-symmetry fixes every point of Ak. ∈ 26 B.-Y. CHEN

S´anchez also proved the following two results in [70].

Theorem 11.3. For each complex ﬂag manifold MC, one has

#k(MC) = dim H∗(MC, Z2). (11.2) Let G be a compact connected semisimple Lie group. Then the homogeneous spaces one obtains as orbits of G under the adjoint representation on the Lie algebra of G are also called generalized flag manifolds. It is well-known that these manifolds admit Kähler-Einstein metrics. It is also well-known that every generalized flag manifold admits k-symmetric structure. In a similar way, Sánchez also proved the following.

Theorem 11.4. If M is a generalized flag manifold, then any of its k- symmetric structure on M satisfies #k(M)= χ(M). 11.4. k-number and k-symmetric submanifolds. Let φ : M Em be → an isometric embedding of a compact Riemannian manifold into the Eu- m clidean m-space. Denote the normal space of M in E at x by Tx⊥M. If φ satisfies the following three properties: m m (a) For each x M, there is an isometry σx : E E such that k ∈ → σ = idM , σx(x)= x, and σx ⊥ = identity on T M; x |Tx M x⊥ (b) σx(M) M; and ⊂ (c) Let θx = σx M . The collection θx, x M defines on M a Rie- | { ∈ } mannian regular s-structure of order k, then M is called an extrinsic k-symmetric submanifold in [35]. Sánchez proved in [70] the following.

Theorem 11.5. If M Em is an extrinsic k-symmetric submanifold, then ⊂ #k(M)= b(M, Zp) (11.3) for any prime number p 2 which divides k. ≥ 11.5. Index numbers and CW complex structures. A CW complex is made of basic building blocks called cells. An n-dimensional closed cell is the image of an n-dimensional closed ball under an attaching map. An n-dimensional open cell is a topological space that is homeomorphic to the n- dimensional open ball. Closure-finite means that each closed cell is covered by a finite union of open cells. The C stands for “closure-finite” and the W for “weak topology”.

Professor Nagano and I made the following conjecture on two-numbers which was presented the ﬁrst time in my report [14, page 53]. TWO-NUMBERS AND THEIR APPLICATIONS - A SURVEY 27

Conjecture. The two-number #2M is equal to the smallest number of cells that are needed for a CW complex structure on a connected compact symmetric space M.

Using the convexity Theorems of Atiyah [3] and of Guillemin-Sternberg [38] for symplectic manifolds with a hamiltonian torus action and the generalization of Duistermaat [32] for ﬁxed point set of antisymplectic involutions, J. Berndt, S. Console and A. Fino gave in [6] alternative proofs of S´anchez’s results mentioned above. Also, related to our conjecture given above, J. Berndt, S. Console and A. Fino proved the following result in [6] (see also [27]).

Theorem 11.6. The index number #IM is equal to the smallest number of cells that are needed for a CW complex structure on M for each real ﬂag manifold M.

12. Conjectures Let M be a connected compact symmetric space. Professor Nagano and I made the following three conjectures.

Conjecture 1. χ(M) # (M) (mod 2) for every connected compact sym- ≡ 2 metric space M.

Conjecture 2. #2M = dim H(M, Z2) for every connected compact symmetric space M.

Remark 12.1. M. Takeuchi proved that Conjecture 2 is true for all symmetric R-spaces in [79]. On the other hand, for every compact symmetric space we could check both Conjectures 1 and 2 are correct.

Conjecture 3. #2M = the smallest number of cells that are needed for a CW complex structure for every connected compact symmetric space M.

Direct computations show that Conjecture 3 is also true for sphere, real projective space as well as for hermitian symmetric spaces.

Remark 12.2. These 3 conjectures remain open till now.

13. Open problems

The total Betti numbers, b(M; R)= i 0 bi(M; R), of a simply-connected compact symmetric space M satisﬁes (see≥ [14, page 53-54]): P # M b(M; R). (13.1) 2 ≥ Professor T. Nagano asked the following. 28 B.-Y. CHEN

Problem 13.1. b(M; R) < # M = M has 2-torsion? 2 ⇒

A. Borel and J. P. Serre deﬁned the p-rank (p prime), rp(G), of a compact Lie group G as the largest integer h such that G contains the direct product of h cyclic groups of order p. Very little were known for p-rank. However, Borel and Serre in [7] proved the next two results.

(a) rp(G) rk(G) for a connected compact Lie group G; ≥ (b) rp(G) > rk(G) implies that G has p-torsion; A. M. Cohen and G. M. Seita proved, via algebraic method in [25], that for an algebraic group G of exceptional type G2, F4, E6, E7, or E8 over an algebraically closed ﬁeld, the maximal elementary abelian p-subgroup E of G with p = 2 and of maximum rank (i.e., rk(E) = rk(G)) is unique up to 6 conjugacy and of types G2, F4, E7∗, and E8 with rk(E)= rk(G) + 1. Besides these, very little were known for p-ranks of compact Lie groups. In view of [25] and [70], we asked the following.

Problem 13.2. Is it possible to establish a geometric theory to determine p-rank rp(G) for compact Lie groups for prime p> 2?

Remark 13.1. We did this for p = 2 in [23].

A Chevalley group is a linear algebraic group of Lie type over ﬁnite ﬁelds related to a semisimple complex Lie algebra. Such groups were introduced by C. Chevalley in [24]. For the Lie algebras An,Bn,Cn, Dn this gave well known classical groups, but Chevalley’s construction also gave groups associated to the exceptional Lie algebras E6, E7, E8, F4, and G2. The ones of type E6 and of type G2 were constructed by L. E. Dickson [30] in 1901 and [31] in 1905, respectively, known as the Dickson groups. While we were working on two-numbers during the 1980s, Professor Nagano asked the following question.

Problem 13.3. Is it possible to extend our study of 2-numbers to Chevalley groups?

Because I don’t know much about ﬁnite group theory, I don’t know how to answer this question. I do hope that someones were able to solve this problem.

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Department of Mathematics, Michigan State University, East Lansing, Michi- gan 48824–1027, U.S.A. E-mail address: [email protected]